Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 31, 2006, 541–554
PARAMETRIZATIONS OF ALGEBRAIC CURVES
A. F. Beardon and T. W. Ng
Centre for Mathematical Sciences, Wilberforce Road
Cambridge CB3 0WB, England; afb@dpmms.cam.ac.uk
The University of Hong Kong, Department of Mathematics
Pokfulam Road, Hong Kong; ntw@maths.hku.hk
Abstract. We show that if a pair of meromorphic functions parametrize an algebraic curve
then they have a common right factor, and we use this to derive a variety of results on algebraic
curves.
1. Introduction
We show that any parametrization of an affine algebraic curve by meromorphic
functions factorizes in a certain way, and we use this to derive several classical
results about the parametrizations of such curves. The distinction between the
affine curve, the projective curve, and the associated compact Riemann surface,
is not always clear in the literature. Here we start with a careful statement of
the Desingularization Theorem (whose proof is now available in accordance with
modern standards) but, apart from this, we use only basic complex analysis. Some
of our results can be found in the literature on algebraic geometry, but we hope
that this partly expository paper will be of interest to complex analysts.
We use C to denote the complex plane, and P1 and P2 to denote the complex projective spaces of dimension one (the extended complex plane) and two,
respectively. Let P (u, v) be an irreducible complex polynomial in two variables.
Then the affine algebraic curve associated with P is
(1.1)
C = {(u, v) ∈ C × C : P (u, v) = 0},
and (because P is irreducible) C determines P to within a non-zero scalar multiple. Given P we can also form the corresponding homogeneous polynomial Pe, and
hence construct the projective curve Ce in P2 , and a compact Riemann surface
R , which is called the desingularization of C , and which (given P ) is unique up
to a conformal map. We shall use P , Pe , C , Ce and R in this sense throughout the
2000 Mathematics Subject Classification: Primary 14H50, 30F10.
The research of the second author was partially supported by a seed funding grant of HKU
and RGC grant HKU 7020/03P.
542
A. F. Beardon and T. W. Ng
paper and without further discussion (the details and formal definitions appear in
Section 2).
The triple (f, g, D) is a meromorphic parametrization of an affine curve C
on the domain D if
(i)
(ii)
(iii)
(iv)
D is a simply connected subdomain of P1 ;
f and g are non-constant and meromorphic in D ;
if f (z) 6= ∞ and g(z) 6= ∞ , then f (z), g(z) ∈ C ;
with finitely many exceptions, every point of C is of the form f (z), g(z)
for some z in D .
We also say that (f, g) is a
(a) rational parametrization if D = P1 , and f and g are rational maps;
(b) polynomial parametrization if D = C , and f and g are polynomials;
(c) entire parametrization if D = C , and f and g are entire functions.
The parametrization of an affine curve is closely related to the uniformization
of an algebraic curve. For example, in [3] Bers gives the following definition of
uniformization. Let Σ be a set in C × C (for example, an affine algebraic curve).
Suppose that f and g are meromorphic in a domain D in C , and let Φ = (f, g)
and D0 = {z ∈ D : f (z), g(z) 6= ∞} . Then f and g uniformize Σ if
(i) Φ(D0 ) is a dense subset of Σ , and
(ii) there is a discrete group of holomorphic self-maps of D such that Φ(z1 ) =
Φ(z2 ) if and only if z2 = g(z1 ) for some g in G (so that D0 /G can be
identified with Φ(D0 ) ).
Throughout, we use juxtaposition
to denote the composition of maps; for
example, f U (z) = f U (z) . We now state our main result (the terms π , Se and
L in this will be defined shortly).
Theorem 1. Suppose that (f, g, D) is a meromorphic parametrization of an
affine algebraic curve. Then f = U h and g = V h for some h , U and V , where
(a) h: D → R is holomorphic and non-constant;
(b) U and V are non-constant and meromorphic on R ;
e
(c) (U, V ): R → P1 ×P1 is injective on R\E where E is the finite set π −1 (S∪L)
.
Theorem 1 implies that the following diagram commutes:
h
D
(U,V )
(f,g)
C
/R
I
/ P1 × P 1 ,
where I is the identity (or inclusion) map. We shall use Theorem 1 to prove the
following known results.
Parametrizations of algebraic curves
543
Theorem 2. If C has a meromorphic parametrization on C then R has
genus zero or one.
Theorem 3. Suppose that C has an entire parametrization (f, g) . Then R
has genus zero and, in Theorem 1, we may assume that R = P1 , U and V are
rational, and h is entire.
Theorem 4. An affine curve C has a rational parametrization if and only if
R has genus zero. If this is so then, in Theorem 1, we may assume that R = P1 ,
and U and V are rational.
Theorem 5. Suppose that an affine algebraic curve C has a rational parametrization. Then it has a polynomial parametrization if and only if it has exactly
one place at infinity. If this is so then, in Theorem 1, we may assume that R = P1 ,
and U and V are polynomials.
Theorem 6. An algebraic curve of genus g can be parametrized by rational
functions if g = 0 , by elliptic functions if g = 1 , and by functions that are
automorphic with respect to some Fuchsian group if g > 2 .
Theorem 7. Suppose that (f, g, D) is a meromorphic parametrization of an
affine algebraic curve, let Γf be the group of conformal automorphisms γ of D
such that f γ = f , and similarly for Γg . Then there exists a positive integer N
with the property that, for every γ in Γf , there is an integer n with 0 6 n 6 N
and γ n ∈ Γg . A similar statement holds with f and g interchanged.
We give some examples to illustrate these theorems. First,
2z
1 − z2
(f, g) =
,
1 + z2 1 + z2
is a rational parametrization of the algebraic curve C given by u2 + v 2 = 1 so, by
Theorem 4, R has genus zero (and so, from the theory of Riemann surfaces, it is
conformally equivalent to P1 ). Now (f, g) = (sin z, cos z) is an entire parametrization of C , and this illustrates Theorem 3 with h(z) = eiz . Note that f and g
have the same set of periods. On the other
hand, the curve given by u2 = v 3 has
an entire parametrization e3πiz , e2πiz where these two parametrizing functions
have different, but common, periods (see Theorem 7). If ℘ is the Weierstrass
elliptic function (with respect to some lattice), then ℘0 (z)2 is a cubic polynomial
in ℘(z) , and this shows that the genus one case can arise in Theorem 2.
Theorem 2 was proved by Picard [13], but see also [7], [10], [11] (where a proof
based on Nevanlinna theory can be found on p. 232), and (for a historical comment)
[5, p. 16]. Theorem 3 is stated in [8], where only a sketch of a proof is offered.
Theorem 4 (due to Lüroth) is proved algebraically in [16, p. 151]; Theorem 5 is
stated without proof in [1]. Theorem 6 is classical, and the statements here are not
reversible in the sense that every algebraic curve can be uniformized by functions
544
A. F. Beardon and T. W. Ng
defined in the upper-half-plane {x + iy : y > 0} (see [3, p. 259]). Theorem 7 is a
generalization of Theorem 1, [15]. We conjecture that Γf ∩ Γg is of finite index in
Γf and Γg , but we are unable to prove this. It is easy to verify this when D = C
and Γf and Γg are lattices.
2. Preliminary results
An irreducible complex polynomial P (u, v) gives rise to the affine curve C
in C × C given in (1.1). If we write the equation P (u, v) = 0 in homogeneous
e v, w) = 0 which gives rise to the projective
co-ordinates we obtain an equation P(u,
curve
e v, w) = 0},
Ce = {[u, v, w] ∈ P2 : P(u,
where we use the notation [u, v, w] for points in projective space P2 . We shall
denote the line at infinity in P2 by L ; this is given by w = 0 . Then Ce can be
expressed as the disjoint union Ce = (Ce \L) ∪ (Ce ∩ L) , where Ce ∩ L is finite, and
the map
(2.1)
α: C → Ce \L,
(u, v) 7→ [u, v, 1],
e , respecis bijective. Let S and Se denote the set of singular points in C and C
tively. Then S and Se are finite sets, and if we give C \S and Ce \Se the usual (and
natural) conformal structure that is described in the general theory of algebraic
curves, we find that the restriction
(2.2)
α: C \S → Ce \(Se ∪ L)
is biholomorphic.
The proofs of Theorems 2–7 are based on the construction of R from the
polynomial P via the affine and projective curves C and Ce . The distinction
between these spaces is often blurred in the literature, but in any rigorous argument
it is essential to distinguish carefully between them. The fact is that C is not
compact, and to remedy this we embed C in the projective curve Ce which is
compact. However, Ce need not be a Riemann surface. To overcome this, we
construct the Riemann surface R (without any direct reference to C or Ce ) as
the space of germs derived from the polynomial P . The link between R and Ce
is then given by the Desingularization Theorem which is stated below. Explicitly,
R is constructed as follows. A function element is a pair (f, D) , where f is
meromorphic in D , and two function elements (f, D) and (g, ∆) are equivalent
at a point ζ in D ∩ ∆ if f = g near ζ . A germ at ζ is an equivalence class of
function elements (f, D) , where ζ ∈ D , and we denote this germ by [f ]ζ . Finally,
R is (essentially) the space of germs [w]ζ as ζ varies, where the function w(z)
satisfies P z, w(z) = 0 in some neighbourhood of ζ .
Parametrizations of algebraic curves
545
We can summarize all this (informally) by saying that whereas R is the space
of germs, the affine curve C is the set of points (z, w) , where w is the value of
the germ at z . As there may be different germs which take the same value at a
given point, we see that R and C are essentially different objects. Consider, for
example, the affine curve C given by v 2 = u3 + u2 , and let
σ(z) =
√
1 + z = 1 + z/2 + · · ·
(which is single-valued and holomorphic in the unit disc D ). Let w1 (z)
= zσ(z)
and w2 (z) = −zσ(z) ; then for all z in D , P z, w1 (z) =
P
z,
w
(z)
= 0 . As
2
w1 (0) = 0 = w2 (0) , we see that 0, w1 (0) and 0, w2 (0) are the same point of
C , namely (0, 0) . However, as w1 and w2 have different Taylor expansions at
the origin, the germs [w1 ]0 and [w2 ]0 are distinct points of R . In conclusion, the
distinct points [w1 ]0 and [w2 ]0 of R correspond to the single point (0, 0) in C ; see
Figure 1. Note (see Theorems 4 and 5) that C has a polynomial parametrization,
namely
u = f (z) = 2z + z 2 , v = g(z) = 2z + 3z 2 + z 3 ,
so that in this case, R is conformally equivalent to P1 .
the affine curve
the Riemann surface
Figure 1.
Every compact surface has a genus, and it is known that if a Riemann surface
has genus zero, then it is conformally equivalent to P1 . The genus of the curve
C , and of Ce , is defined to be the genus of the corresponding Riemann surface R .
Thus if C has genus zero, then we may assume that R = P1 . We shall need other
results on Riemann surfaces (for example, the Uniformization Theorem and the
Riemann–Hurwitz Formula) as well as the basic theory of algebraic curves, and
we refer the reader to [2], [4], [6], [9], [12], [16] and [17] for more details.
Recall that S and Se are the sets of singular points of C and Ce , respectively.
The link between R and Ce is given in the following fundamental result (modern
proofs of which can be found in, for example, [6, p. 169], [9, pp. 66–82] and [12,
p. 192]).
546
A. F. Beardon and T. W. Ng
The Desingularization Theorem I. Given C , there exists a compact Riee is a finite
mann surface R , and a surjective map π: R → Ce such that π −1 (S)
−1 e
e
set, and π: R\π (S) → C \Se is biholomorphic.
The compact Riemann surface R is called the desingularization or normalization of the projective curve Ce , and it is unique up to a conformal map.
As Theorem 1 is concerned with maps from R into C × C (instead of maps
into P2 ), we want to convert the map π: R → Ce into a map β: R → C and
study this. To do this we recall that Ce ∩ L is finite. As π is holomorphic on the
compact surface R , we see that π −1 (L) is a finite subset of R . Now let
e ∪ π −1 (L) = π −1 (Se ∪ L).
E = π −1 (S)
It follows that E is a finite subset of R and, from (2.1), that the composition
α−1
π
e ∪ L −→
(2.3)
β: R\E −→ Ce \ S
C \S
is biholomorphic. As C ⊂ C × C , we can write β = (U, V ) , where U : R\E → C
and V : R\E → C are holomorphic. This yields the following corollary.
The Desingularization Theorem II. Given C , there exists a compact
Riemann surface R , and a biholomorphic map β = (U, V ) of R\E onto C \S ,
where E = π −1 (Se ∪ L) is a finite subset of R .
According to [16, p. 96], the concept of a place is the algebraic counterpart
of a branch of a curve over C . We say that Ce (or C ) has one place at infinity
if π −1 (Ce ∩ L) contains exactly one point. In other words, Ce has one place at
infinity if it has exactly one point on the line at infinity, and if this point is the
image under π of exactly one point of R . For example, the affine curve given by
w 2 = z 4 − 3z 3 has two places at infinity (see [14]) and so cannot be parametrized
by polynomials. In this case, Ce ∩ L is the single point [0, 1, 0] , and the two places
arise from the two sets of germs {[w1 ]z : |z| > 3} and {[w2 ]z : |z| > 3} , where
w1 (z) = z(1 − 3/z)1/2 and w2 (z) = −z(1 − 3/z)1/2 , and where (1 − 3/z)1/2 is the
single-valued function on |z| > 3 that takes the value 1 at ∞ . By contrast, the
affine curve given by (8z + 1)z 2 = 9w 2 has a polynomial parametrization, namely
(f, g) , where
f (z) = 12 z(z + 1), g(z) = 61 z(z + 1)(2z + 1).
The corresponding homogeneous curve is given by (8u + w)u2 = 9v 2 w , and this
meets L at the single point [0, 1, 0] . For sufficiently large |u| , the germs of v (as
a function of u ; equivalently, the points on R ) are given by
√ p
v(u) = ± u u 8 + 1/u /3,
p
8 + 1/u that
where we are taking
(for
|u|
>
1/8
)
the
single
valued
choice
of
√
takes the value 2 2 at ∞ . As these germs are converted into each other by
analytic continuation around ∞ we see (as we already knew from Theorem 5)
that this curve has only one place at infinity.
Parametrizations of algebraic curves
547
3. The proof of Theorem 1
The idea behind our proof is to consider the maps
f, g: D → P1 ,
U, V : R → P1 ,
−1
ϕ = (f, g): D → C ,
β = (U, V ) : R → C ,
ϕ: D → R,
π: R → Ce ,
h=β
which act as indicated in the following commutative diagram except that in each
case there may be some finite exceptional set on which the function may not yet
be defined.
C
}> `AAA β
}
AA
}}
AA
}}
A
}
}
h
/R
D@
@@
~
~
@@
~~
@@
~
f,g
@ ~~~~ U,V
P1
ϕ
π
/ e
C.
Our proof depends on giving a complete description of what is happening at
these exceptional points, and for this we must show that certain isolated singularities of a meromorphic function are removable. It is known that any isolated
singularity of an injective analytic map is removable, and we shall need the following mild extension of this result.
Lemma 8. Suppose that X is a finite subset of a Riemann surface R , and
that F : R\X → C is holomorphic. Suppose also that there is an integer M such
that for all a in R\X , the equation F (z) = F (a) has (counting multiplicities) at
most M solutions in R\X . Then F extends to a meromorphic function on R .
Proof. We may assume that M is minimal; thus there is some a in R\X
such that the solutions of F (z) = F (a)P
are, say, a1 , . . . , aq , where a1 = a , and
where the valency of F at aj is kj , and j kj = M . It is well known that we can,
for each j , construct mutually disjoint open neighbourhoods Nj of aj so that Nj
lies in a parametric disc at aj , and such that the restriction of FTto Nj is, up to
a conformal change of coordinates, the map z 7→ z kj . Let N = j F (Nj ) . Then
N is an open neighbourhood
of F (a) , and becauseSeach point in N has exactly
S
M pre-images in j Nj , it follows that F −1 (N ) ⊂ j Nj .
Now choose a point ζ in X . We may assume that the closures of the Nj
chosen above do
S not contain ζ , so we can find a neighbourhood N of ζ that is
disjoint from j Nj , and hence that F (N ) ∩ N = ∅ . We may assume that N
548
A. F. Beardon and T. W. Ng
lies in a parametric disc at ζ , and if we consider the restriction of F to N \{ζ} ,
and then apply the Weierstrass–Casorati Theorem, we see that ζ is a removable
singularity of F .
We now prove Theorem 1.
The proof of Theorem 1. First, define ϕ: D → P1 × P1 by ϕ(z) = f (z), g(z) .
Next, let L be the set of (u, v) in P1 × P1 such that u = ∞ or v = ∞ . If
z ∈ D\ϕ−1 (S ∪ L) then f (z) 6= ∞ , g(z) 6= ∞ and f (z), g(z) is a non-singular
point of C ; thus ϕ maps D\ϕ−1 (S ∪ L) into C \S . We let K = ϕ−1 (S ∪ L) ; then,
as S is finite, and the poles of f and g are isolated, we see that K is a discrete
subset of D .
Next, we define a holomorphic mapping h: D\K → R\E as the composition
of maps given by
ϕ
β −1
h: D\K −→ C \S −→ R\E.
Also, as h = β −1 ϕ we have
(3.1)
(f, g) = ϕ = βh = (U h, V h)
on D\K , where β = (U, V ) , and U and V are maps of R\E into C . These
maps are illustrated in the following commutative diagram in which
(1) α , β and π are biholomorphic, and
(2) h , U and V are holomorphic;
see (2.2) and (2.3).
α
/ Ce \(Se ∪ L)
C \S
<y
bEE
9
EE β
tt
ϕ yyy
t
E
EE
y
tt
EE
yy
tt π
y
t
y
t
h
/ R\E U,V
/ C.
D\K
Our first task is to show that U and V are meromorphic on R ; that is,
that U and V are holomorphic maps of R onto P1 . Note that once U and V
are holomorphic on R they are (as maps between compact Riemann surfaces)
necessarily surjective. Let card (X) denote the cardinality of a set X . Now take
any a in R\E . As β is biholomorphic, and β = (U, V ) , we have
card {z ∈ R\E : U (z) = U (a)} = card β{z ∈ R\E : U (z) = U (a)}
6 card {(u, v) ∈ C : u = U (a)}
= card
U (a), v) ∈ C
6 d,
Parametrizations of algebraic curves
549
where d is the degree of v in P (u, v) . Thus Lemma 8 is applicable, and U extends
to a meromorphic function on R . Clearly the same argument applies to V . As
β is biholomorphic on R\E , Theorem 1(b) and Theorem 1(c) hold.
It remains to show that Theorem 1(a) holds. Now h = β −1 ϕ , β is biholomorphic, and ϕ = (f, g) is not constant, so that h is not constant. As h is
holomorphic on D\K , where K is a discrete subset of D , it remains to show that
the (isolated) points of K are removable singularities of h .
Select any z0 in K . Now f (z0 ) ∈ P1 so that U −1 f (z0 ) ⊂ R . We have
seen above that U has degree
at most d , so that we can let w1 , . . . , wq be the
−1
distinct points of U
f (z0 ) , where q 6 d . Now choose a positive r such that the
compact neighbourhood N = {z : |z − z0 | 6 r} of z0 has the following properties:
(a) N does not contain any pointof K other than z0 ;
(b) the components of U −1 f (N ) are Nj , j = 1, . . . , q , where, for each j , Nj
is a compact neighbourhood of wj ;
(c) each Nj lies in some co-ordinate chart of R .
Note that
(d) h is holomorphic in N0 = {z : 0 < |z − z0 | < r} , and
(e) the Nj are pairwise disjoint and compact.
As (f, g) = (U h, V h) on D\K we see that f (N0 ) = U h(N0 ) , so that
S
h(N0 ) ⊂ U −1 f (N0 ) = Nj .
j
As h(N0 ) is connected, it lies in some Nj , and it follows from (c) above that z0
is a removable singularity of h . The proof of Theorem 1 is complete.
4. The proof of Theorem 2
The proof mimics the usual proof of Picard’s Little Theorem. Let (f, g) be
a meromorphic parametrization of C on C . Then, by Theorem 1, there exists
a non-constant holomorphic map h: C → R . Now let Rb be the universal cover
of R . As C is simply connected, we can lift h to a non-constant holomorphic
map ĥ: C → Rb . Liouville’s Theorem implies that Rb cannot be the unit disc;
thus Rb is either C or P1 , so that R has genus zero or one.
5. The proof of Theorem 3
By Theorem 2, R is of genus zero or one. First, we suppose that R is of
genus one and reach a contradiction. As R has genus one the universal covering
space of R is C , and we let π0 : C → R be a universal covering map. By
Theorem 1, there exists a non-constant holomorphic map h: C → R and nonconstant meromorphic maps U and V on R such that (f, g) = (U h, V h) on C .
Now lift h to a holomorphic map h0 : C → C such that π0 h0 = h . As R is
compact, U and V are surjective, so there is some point a with U (a) = ∞ . Now
550
A. F. Beardon and T. W. Ng
h(z) 6= a in C , for otherwise there is some z with f (z) = U h(z) = U (a) = ∞ , and
this contradicts the fact that f is entire. As π0 h0 = h , it follows that h0 cannot
take any value in the infinite set π0−1 ({a}) , and this contradicts Picard’s Little
Theorem. Therefore, R has genus zero and so we may assume that R = P1 . It
follows from this that U and V are rational functions.
Finally, we must show that h can be taken to be entire. If a = ∞ , then
(because h 6= a ) h is entire. If a 6= ∞ , let L(z) = 1/(z − a) and U1 = U L−1 ,
V1 = V L−1 and h1 = Lh . Then f = U1 h1 , g = V1 h1 , U1 and V1 are rational
and h1 is entire.
6. The proof of Theorem 4
(i) Suppose first that R has genus zero; thus, by applying a conformal map,
we may assume that R = P1 . Then U and V are meromorphic maps from P1 to
itself and so are rational. In this case, (U, V ) is a rational parametrization of C .
(ii) Now suppose that (f, g) is a rational parametrization of C . Then, in
Theorem 1, D = P1 so there exists a non-constant holomorphic map h: P1 → R ,
and non-constant meromorphic maps U : R → P1 , and V : R → P1 , such that
(f, g) = (U h, V h) . The Riemann–Hurwitz Formula applied to h implies that the
genus of R is at most the genus of P1 , so that R has genus zero. We may now
assume that R = P1 , and this implies that U and V are rational functions.
7. The proof of Theorem 5
Throughout this proof we consider an affine algebraic curve C with a rational
parametrization. We have to show that C has a polynomial parametrization if and
only if π −1 (L) contains exactly one point of R . We must also show that, under
these circumstances, U and V may be taken to be polynomials. Theorem 4 implies
that we may take R = P1 , and that there
exist rational functions U and V , and
a finite set E , such that P U (z), V (z) = 0 on C\E , and β(z) = U (z), V (z)
e , where S and Se are the sets of
is injective on C\E . We recall that S ⊂ α−1 (S)
singular points of C and Ce , respectively.
(i) First we show that if C has only one place at ∞ then it has a polynomial
parametrization. Let P be the set of poles of U or V . As U and V are rational,
P is a finite non-empty set. If P = {∞} , then U and V are polynomials and
(U, V ) is a polynomial parametrization of the curve. Thus we may now assume
that P contains a point z0 in C . We are now going to show that P = {z0 } .
As z0 ∈ P , we can write
U1 (z)
V1 (z)
U (z) =
, V (z) =
,
m
(z − z0 )
(z − z0 )n
where U1 , V1 are rational functions, U1 (z0 ) and V1 (z0 ) are nonzero, m > 0 ,
n > 0 , and either m > 0 or n > 0 . Without loss of generality, we may assume
that m > n , m > 1 and n > 0 .
551
Parametrizations of algebraic curves
Now consider the following mapping diagram where α and β are injective
and holomorphic, and π is biholomorphic.
e ∪E
C\ β −1 α−1 (S)
Φ
π
β
/ Ce \S.
e
e
C \α−1 (S)
/ R\π −1 (S)
e
α
Let Φ = π −1 αβ ; clearly, this is injective and analytic except possibly at
isolated points. Also,
lim Φ(z) = lim π −1 αβ(z)
z→z0
z→z0
= lim π −1 α U (z), V (z)
z→z0
= lim π
z→z0
−1
V1 (z)
U1 (z)
,
,1
m
(z − z0 ) (z − z0 )n
= lim π −1 U1 (z), V1 (z)(z − z0 )m−n , (z − z0 )m
z→z0
= π −1 [a, b, 0] ,
for some complex numbers a , b (not both zero). This shows that z0 is a removable
singularity of Φ . Moreover, since Ce is closed, [a, b, 0] ∈ Ce ∩ L , and as π −1 (Ce ∩ L)
contains exactly one point, say p , we must have Φ(z0 ) = p . Clearly, this argument
holds for any z0 in P ∩ C . However, as Φ is holomorphic and injective on the
complement of a finite set, any holomorphic extension to isolated points must also
lead to an injective map. It follows that if z1 ∈ P ∩ C , then Φ(z1 ) = p so that
z1 = z0 . We conclude that P ∩ C = {z0 } and, consequently, that
U (z) =
U1 (z)
,
(z − z0 )m
V (z) =
V1 (z)
,
(z − z0 )n
where U1 and V1 are polynomials with U1 (z0 ) 6= 0 , V1 (z0 ) 6= 0 , m > 1 and
n > 0.
We now claim that deg(U1 ) 6 m and deg(V1 ) 6 n (equivalently, that U (∞)
and V (∞) are finite). We assume the contrary and let d = max{m, n} and
k = max{deg(U1 ) − m + d, deg(V1 ) − n + d, d}.
552
A. F. Beardon and T. W. Ng
Then k > d . Note that as z → ∞ ,
V1 (z)
U1 (z)
,
, 1 = U1 (z)(z − z0 )d−m , V1 (z)(z − z0 )d−n , (z − z0 )d
m
n
(z − z0 ) (z − z0 )
U1 (z)(z − z0 )d−m V1 (z)(z − z0 )d−n (z − z0 )d
=
,
,
zk
zk
zk
→ [c0 , d0 , 0],
where c0 and d0 are complex numbers, not both zero. As before we get Φ(∞) = p ,
which is again a contradiction. Thus, finally, U1 and V1 are polynomials with
deg(U1 ) 6 m and
let s = 1/(z − z0 ) , then both U (z) =
deg(V1 ) 6 n . Now
−1
m
−1
U1 (s + z0 )s
and V (z) = V1 (s + z0 )sn are polynomials in s and we have
obtained a polynomial parametrization of C .
(ii) We show that if C has a polynomial parametrization then it has only one place
at ∞ . Let (f, g) be a polynomial parametrization of C . Then, by Theorem 3,
we may assume that R = P1 , and that f = U h , g = V h , where U , V are
rational, and h is a polynomial. Since f is a polynomial, f −1 (∞) = {∞} . Hence
h−1 U −1 (∞) = {∞} . It follows that U −1 (∞) = {h(∞)} = {∞} and therefore U
is a polynomial. Similarly, V is also a polynomial.
Now
Φ = π −1 αβ: P1 \E1 → P1 \E2
is injective, where E1 and E2 are some finite sets, and it follows from this that
any point in E1 is a removable singularity of Φ . Thus we can extend Φ to the
whole of P1 , and this extension will be a map of P1 onto itself. Next, πΦ = αβ
on the dense subset C\E1 of C . Since both πΦ and αβ are continuous on C ,
we now see that πΦ = αβ on C .
Now suppose that π −1 (Ce ∩ L) contains two different points p and q . Then
there are distinct points p0 and q0 in P1 such that Φ(p0 ) = p and Φ(q0 ) = q .
One of them, say p0 , is in C , and hence αβ(p0 ) ∈ α(C ) . On the other hand,
πΦ(p0 ) ∈ Ce ∩ L , and this contradicts the fact that πΦ = αβ on C . The proof is
complete.
8. The proof of Theorem 6
We have seen that if R is of genus zero, then we have a rational parametrization of C . If R is of genus one then, by composing U and V with the universal
covering map of R , we conclude that C can be parametrized by elliptic functions.
Now suppose that R has genus greater than one. Then, by the Uniformization
Theorem, R = D/Γ for some Fuchsian group Γ acting on the unit disc D without
elliptic elements. Each point of R is then a Γ -orbit [z]Γ of a point z in D . Now
define U1 and V1 by U1 (z) = U ([z]Γ ) and V1 (z) = V ([z]Γ ) . Then U1 and V1 are
automorphic functions invariant under Γ , and (U1 , V1 ) parametrizes C .
Parametrizations of algebraic curves
553
9. The proof of Theorem 7
We suppose that (f, g, D) is a meromorphic parametrization of P (u, v) = 0 ,
and we write
P (u, v) = a0 (u) + a1 (u)v + · · · + am (u)v m ,
where am (z) is not identically zero. Choose any complex number z0 in D such
that the orbit Γf (z0 ) does not contain a pole of
f or g , or a zero of am (u) . Then
the polynomial p defined by p(t) = P f (z0 ), t is not constant and, for each γ in
Γf , and each n = 1, 2, . . . , m + 1 ,
p gγ n (z0 ) = P f γ n (z0 ), gγ n (z0 ) = 0.
As p has exactly m zeros, there must be some distinct s and t (which may
depend on z0 ) such that 1 6 s < t 6 m + 1 and gγ s (z0 ) = gγ t (z0 ) . As there is an
uncountable number of choices of z0 here, there must be an uncountable set U of
z for which the integers s and t are independent of z0 in U . As any uncountable
subset of C (for example, U ) has an uncountable set of accumulation points, we
deduce that gγ s (z) = gγ t (z) for all z in D or, equivalently, that γ s−t ∈ Γg . This
completes the proof.
Acknowledgements. The authors would like to thank Y. K. Lau and X. Sun
for the helpful suggestions.
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Received 7 December 2005